Median
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Median[data]
gives the median estimate of the elements in data.
Median[dist]
gives the median of the distribution dist.
Details
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- Median is a robust location estimator, which means it is not very sensitive to outliers.
- For VectorQ data
, the median can be thought of as the "middle value".
- Formally, when data is sorted as
, the median is given by center element
if
is odd and the mean of the two center elements
if
is even.
- Median[data] is equivalent to Quantile[data,1/2,{{1/2,0},{0,1}}].
- For MatrixQ data, the median is computed for each column vector with Median[{{x1,y1,…},{x2,y2,…},…}] equivalent to {Median[{x1,x2,…}],Median[{y1,y2,…}],…}. »
- For ArrayQ data, median is equivalent to ArrayReduce[Median,data,1]. »
- The data can have the following additional forms and interpretations:
-
Association the values (the keys are ignored) » SparseArray as an array, equivalent to Normal[data] » QuantityArray quantities as an array » WeightedData based on the underlying EmpiricalDistribution » EventData based on the underlying SurvivalDistribution » TimeSeries, TemporalData, … vector or array of values (the time stamps ignored) » Image,Image3D RGB channels values or grayscale intensity value » Audio amplitude values of all channels » DateObject, TimeObject list of dates or list of times » - Median[dist] is the minimum of the set of number(s) m such that Probability[x≤m,xdist]≥1/2 and Probability[x≥m,xdist]≥1/2. »
- For a continuous distribution dist, the median can be defined using the cumulative distribution function:
.
- Median[dist] is equivalent to Quantile[dist,1/2].
- For a random process proc, the median function
can be computed for slice distribution at time t, SliceDistribution[proc,t], as Median[SliceDistribution[proc,t]]. »
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Examples
open allclose allBasic Examples (4)
Scope (24)
Basic Uses (8)
Exact input yields exact output:
Approximate input yields approximate output:
Find the median of WeightedData:
Find the median of EventData:
Find the median of TemporalData:
Find the median of a TimeSeries:
The median depends only on the values:
Array Data (5)
Median for a matrix gives columnwise medians:
Median for a tensor gives columnwise medians at the first level:
When the input is an Association, Median works on its values:
SparseArray data can be used just like dense arrays:
Find median of a QuantityArray:
Image and Audio Data (2)
Date and Time (5)
Applications (7)
The median represents the center of a distribution:
The median for a distribution without a single mode:
Find the median length, in miles, for 141 major rivers in North America:
Plot a Histogram for the data:
Probability that the length exceeds 90% of the median:
Smooth an irregularly spaced time series using a moving median:
Obtain a robust estimate of location when outliers are present:
Extreme values have a large influence on the Mean:
Compute medians for slices of a collection of paths of a random process:
Properties & Relations (7)
Median is equivalent to a parametrized Quantile:
For nearly symmetric samples, Median and Mean are nearly the same:
For univariate data, Median coincides with SpatialMedian:
The Median of absolute deviations from the Median is MedianDeviation:
MovingMedian is a sequence of medians:
For any distribution, there is InverseCDF[dist,1/2]=Median[dist]:
Similarly for InverseSurvivalFunction:
For a continuous distribution, there is CDF[dist,Median[dist]]=1/2:
Similarly for SurvivalFunction:
Possible Issues (2)
Neat Examples (1)
The distribution of Median estimates for 20, 100, and 300 samples:
Text
Wolfram Research (2003), Median, Wolfram Language function, https://reference.wolfram.com/language/ref/Median.html (updated 2024).
CMS
Wolfram Language. 2003. "Median." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/Median.html.
APA
Wolfram Language. (2003). Median. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Median.html